Linear dependence of multivariable functions It is well known that the Wronskian is a great tool for checking the linear dependence between a set of functions of one variable.
Is there a similar way of checking linear dependance between two functions of two variables (e.g. $P(x,y),Q(x,y)$)?
Thanks.
 A: See http://en.wikipedia.org/wiki/Wronskian and in particular the section "Generalized Wronskian".
A: For checking linear dependency between two functions of two variables we can follow the follwing theorem given by "Green, G. M., Trans. Amer. Math. Soc., New York, 17, 1916,(483-516)". 
Theorem: Let $y_{1}$ and $y_{2}$ be functions of two independence variables $x_{1}$ and $x_{2}$ i.e., $y_{1} = y_{1}(x_{1} ,x_{2}) $ and $y_{2} = y_{1}(x_{1} ,x_{2}) $ for which all partial derivatives of $1^{st}$ order, $\frac{\partial y_{1}}{\partial x_{k}}$, $\frac{\partial y_{2}}{\partial x_{k}}$, $(k = 1,2)$ exists throughout the region $A$. Suppose, farther, that one of the functions, say $y_{1}$, vanishes at no point of $A$. Then if all the two rowed determinants in the matrix
\begin{pmatrix} 
y_{1} & y_{2} \\
\frac{\partial y_{1}}{\partial x_{1}} & \frac{\partial y_{2}}{\partial x_{1}} \\
\frac{\partial y_{1}}{\partial x_{2}} & \frac{\partial y_{2}}{\partial x_{2}}
\end{pmatrix}
vanish identically in $A$, $y_{1}$ and $y_{2}$ are linearly dependent in $A$, and in fact $y_{2}=c y_{1}$.
